It's possible that you have the c spine issue plus the exercise + desk work compounding things.

Have you ever had any repetitive motion injuries? or RSIs?
carpal tunnel, elbow pain, neck pain, shoulder pain, wrist pain?

how is your posture over all & at the desk?
hunched/slouched/ head forward?

If any of those things fit you may want to read on our thoracic outlet syndrome (TOS) forum - the useful sticky thread will be a good place to start-
http://neurotalk.psychcentral.com/thread84.html

even if it doesn't fit - there are many therapies mentioned also


As far as push ups are concerned - the head weighs about 8-14 lbs and with a c spine problem already - putting that much extra weight out for the neck to hold up against gravity is a bit much I think...